Tohoku Mathematical Journal
2020
December
SECOND SERIES VOL. 72, NO. 4

Tohoku Math. J.
72 (2020), 527-535

Title GENERALIZED CARTAN-BEHNKE-STEIN'S THEOREM AND $Q$-PSEUDOCONVEXITY IN A STEIN MANIFOLD

Author Shun Sugiyama

(Received July 2, 2018, revised April 25, 2019)
Abstract. Let $D$ be an open subset of an $n$-dimensional Stein manifold, where $n\geq 2$. Assume that the canonical map $H^{n-1}(D,\mathcal{O})\to H^{n-1}(D,\mathcal{M})$ is injective. Then, we prove that $D$ is pseudoconvex of order 1, which generalizes the well-known theorem of Cartan-Behnke-Stein.

Mathematics Subject Classification. Primary 32F10; Secondary 32E10, 32F32.
Key words and phrases. Stein manifold, $q$-pseudoconvexity, Cartan-Behnke-Stein's theorem.

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